8.2 Some formal considerations

Before we proceed with a more involved example, let us formalize the procedure outlined above. We mainly follow the excellent treatment given in [124], although we restrict ourselves to the cases where is a finite regular subalgebra of .

In the previous example, we performed the decomposition of the roots (and the ensuing decomposition of the algebra) with respect to one of the simple roots which then defined the level. In general, one may consider a similar decomposition of the roots of a rank Kac–Moody algebra with respect to an arbitrary number of the simple roots and then the level is generalized to the “multilevel” .

We consider a Kac–Moody algebra of rank and we let be a finite regular rank subalgebra of whose Dynkin diagram is obtained by deleting a set of nodes from the Dynkin diagram of .

Let be a root of ,

To this decomposition of the roots corresponds a decomposition of the algebra, which is called a gradation of and which can be written formally as
where for a given , is the subspace spanned by all the vectors with that definite value of the multilevel,
Of course, if is finite-dimensional this sum terminates for some finite level, as in Equation (8.11) for . However, in the following we shall mainly be interested in cases where Equation (8.13) is an infinite sum.

We note for further reference that the following structure is inherited from the gradation:

This implies that for we have
which means that is a representation of under the adjoint action. Furthermore, is a subalgebra. Now, the algebra is a subalgebra of and hence we also have
so that the subspaces at definite values of the multilevel are invariant subspaces under the adjoint action of . In other words, the action of on does not change the coefficients .

At level zero, , the representation of the subalgebra in the subspace contains the adjoint representation of , just as in the case of discussed in Section 8.1. All positive and negative roots of are relevant. Level zero contains in addition singlets for each of the Cartan generator associated to the set .

Whenever one of the ’s is positive, all the other ones must be non-negative for the subspace to be nontrivial and only positive roots appear at that value of the multilevel.

8.2.2 Weights of and weights of

Let be the module of a representation of and be one of the weights occurring in the representation. We define the action of in the representation on as

(we consider representations of for which one can speak of “weights” [116]). Any representation of is also a representation of . When restricted to the Cartan subalgebra of , defines a weight , which one can realize geometrically as follows.

The dual space may be viewed as the -dimensional subspace of spanned by the simple roots , . The metric induced on that subspace is positive definite since is finite-dimensional. This implies, since we assume that the metric on is nondegenerate, that can be decomposed as the direct sum

To that decomposition corresponds the decomposition
of any weight, where and . Now, let (). One has because : The component perpendicular to drops out. Indeed, for .

It follows that one can identify the weight with the orthogonal projection of on . This is true, in particular, for the fundamental weights . The fundamental weights project on for and project on the fundamental weights of the subalgebra for . These are also denoted . For a general weight, one has

and
The coefficients can easily be extracted by taking the scalar product with the simple roots,
a formula that reduces to
in the simply-laced case. Note that even when is non-spacelike.

8.2.3 Outer multiplicity

There is an interesting relationship between root multiplicities in the Kac–Moody algebra and weight multiplicites of the corresponding -weights, which we will explore here.

For finite Lie algebras, the roots always come with multiplicity one. This is in fact true also for the real roots of indefinite Kac–Moody algebras. However, as pointed out in Section 4, the imaginary roots can have arbitrarily large multiplicity. This must therefore be taken into account in the sum (8.13).

Let be a root of . There are two important ingredients:

• The multiplicity of each at level as a root of .
• The multiplicity of the corresponding weight at level as a weight in the representation of . (Note that two distinct roots at the same level project on two distinct -weights, so that given the -weight and the level, one can reconstruct the root.)

It follows that the root multiplicity of is given as a sum over its multiplicities as a weight in the various representations at level . Some representations can appear more than once at each level, and it is therefore convenient to introduce a new measure of multiplicity, called the outer multiplicity , which counts the number of times each representation appears at level . So, for each representation at level we must count the individual weight multiplicities in that representation and also the number of times this representation occurs. The total multiplicity of can then be written as

This simple formula might provide useful information on which representations of are allowed within at a given level. For example, if is a real root of , then it has multiplicity one. This means that in the formula (8.25), only the representations of for which has weight multiplicity equal to one are permitted. The others have . Furthermore, only one of the permitted representations does actually occur and it has necessarily outer multiplicity equal to one, .

The subspaces can now be written explicitly as

where denotes the module of the representation and is the number of inequivalent representations at level . It is understood that if for some and , then is absent from the sum. Note that the superscript labels multiple modules associated to the same representation, e.g., if this contributes to the sum with a term

Finally, we mention that the multiplicity of a root can be computed recursively using the Peterson recursion relation, defined as [116]

where denotes the set of all positive integer linear combinations of the simple roots, i.e., the positive part of the root lattice, and is the Weyl vector (defined in Section 4). The coefficients are defined as
and, following [19], we call this the co-multiplicity. Note that if is not a root, this gives no contribution to the co-multiplicity. Another feature of the co-multiplicity is that even if the multiplicity of some root is zero, the associated co-multiplicity does not necessarily vanish. Taking advantage of the fact that all real roots have multiplicity one it is possible, in principle, to compute recursively the multiplicity of any imaginary root. Since no closed formula exists for the outer multiplicity , one must take a detour via the Peterson relation and Equation (8.25) in order to find the outer multiplicity of each representation at a given level. We give in Table 38 a list of root multiplicities and co-multiplicities of roots of up to height 10.

 Table 38: Multiplicities and co-multiplicities of all roots of up to height 10.
 0 0 1 1 1 2 0 0 0 0 1 0 1 1 2 0 1 0 1 0 0 1 1 2 0 0 0 0 1 1 1 1 2 0 0 1 1 0 1 1 0 2 2 0 3/2 1 0 3 3 0 4/3 1 0 4 4 0 7/4 1 0 5 5 0 6/5 1 0 1 1 1 1 1 0 2 2 2 3/2 1 0 3 3 3 4/3 1 0 1 2 0 1 1 2 2 4 0 1/2 0 8 3 6 0 1/3 0 2 2 1 0 1 1 2 4 2 0 1/2 0 8 6 3 0 1/3 0 18 1 2 1 1 1 0 2 4 2 3/2 1 0 2 1 1 1 1 2 4 2 2 1/2 0 8 1 2 2 1 1 2 2 4 4 1/2 0 8 2 2 1 2 2 -2 4 4 2 8 7 -8 2 3 0 1 1 2 4 6 0 1/2 0 8 3 2 0 1 1 2 6 4 0 1/2 0 8

 2 3 1 2 2 -2 3 2 1 1 1 0 2 4 1 1 1 2 2 3 2 2 2 -2 3 2 2 1 1 2 3 3 1 3 3 -4 3 4 0 1 1 2 4 3 0 1 1 2 2 3 3 1 1 2 3 4 1 3 3 -4 2 4 3 1 1 2 3 3 2 3 3 -4 4 3 1 2 2 -2 3 4 2 5 5 -6 3 5 1 1 1 0 4 3 2 2 2 -2 4 4 1 5 5 -6 4 5 0 1 1 2 5 4 0 1 1 2 3 4 3 3 3 -4 3 5 2 3 3 -4 4 3 3 1 1 2 4 5 1 5 5 -6 5 4 1 3 3 -4